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CMSC 351:Fall 2002 Khuller & Postow
Information CMSC
Asymptotic Notations.
Θ(g(n)) = {f (n): there exist positive constants c 1 , c 2 , and n 0 such that 0 ≤ c 1 g(n) ≤ f (n) ≤ c 2 g(n) for all n ≥ n 0 }.
O(g(n)) = {f (n): there exist positive constants c and n 0 such that 0 ≤ f (n) ≤ cg(n) for all n ≥ n 0 }.
Ω(g(n)) = {f (n): there exist positive constants c and n 0 such that 0 ≤ cg(n) ≤ f (n) for all n ≥ n 0 }.
Logarithms.
a = blogb^ a^ logc(ab) = logc a + logc b logb an^ = n logb a
logb a = logc^ a logc b
logb(1/a) = − logb a logb a = 1 loga b
alogb^ n^ = nlogb^ a
Stirling’s Formula.
n! ≈
( (^) n e
)n √ 2 πn
Recurrences “Master Theorem”:
T (n) =
aT (n/b) + cnd^ n > 1 f n = 1
implies
T (n) =
f + (^) ab−cd− 1
nlogb^ a^ − cn
d ab−d− 1 =
Θ(nlogb^ a) a > bd Θ(nd) a < bd nd(f + c logb n) = Θ(nd^ logb n) a = bd
Facts: Number of leaves in a binary tree of height h is at most 2h.
Summations. Distribution law: (^) ( ∑m
i=
ai
∑^ n
j=
bj
∑^ m
i=
∑^ n
j=
aibj
Interchanging order of summation: ∑m
i=
∑^ n
j=
aij =
∑^ n
j=
∑^ m
i=
aij
Splitting range: ∑n
k=
ak =
∑^ r
k=
ak +
∑^ n
k=r+
ak
Arithmetic series: (^) n ∑
k=
k = 1 + 2 + · · · + n = n(n^ + 1) 2
Geometric series: (^) n ∑
k=
xk^ = 1 + x + +x^2 · · · + xn^ =
xn+1^ − 1 x − 1 x^6 = 1
∑^ ∞
k=
xk^ =
1 − x
x < 1
Another useful sum: (^) ∞ ∑
k=
kxk^ = x (1 − x)^2
x < 1
Harmonic series:
Hn = 1 +
4 +^ · · ·^ +
n =
∑^ n
k=
k = ln^ n^ +^ O(1)
Telescoping series: ∑n
k=
(ak − ak− 1 ) = an − a 0
Products: (^) n ∏
k=
ak = a 1 a 2 · · · an log
∏^ n
k=
ak =
∑^ n
k=
log ak
Quadratic Formula.
ax^2 + bx + c = 0 ⇒ x =
−b ±
b^2 − 4 ac 2 a
